# Question on the spinor Indices, in non-relativistic quantum mechanics

I've caught by a loop of:

Standard texts of Non-Relativistic Quantum Mechanics $$\to$$ Representation theory of Lie groups and Lie algebras of $$SO(3)$$ and $$SU(2)$$ $$\to$$ Discussions of infinitesimal transformations and formulas like $$\Lambda_{\alpha}^{\beta} = \mathrm{exp}\Big(\frac{1}{2}\Omega_{\mu\nu}(M^{\mu\nu})_{\alpha}^{\beta}\Big)$$ and $$S[\Lambda]_{a}^{b} = \mathrm{exp}\Big(\frac{1}{2}\Omega_{\mu\nu}(S^{\mu\nu})_{a}^{b}\Big)$$ $$\to$$ Classical Field theory and Noether theorem $$\to$$ Active/Passive trasformations $$\to$$ Standard texts of Relativistic Quantum Mechanics

But what I really want to understand is simply: what are Pauli spinors! For tensors is simple: they are elements of the tensor product. But for some reason the active/passive views ARE important.

Now, to be clear in this question, consider the Group: $$SO(1,3)^{+}_{\uparrow}$$. I understand completely when someone writes:

$$V^{\mu'} = \Lambda^{\mu'}_{\nu}V^{\nu}, \tag{1}$$

this is a 4-vector. I don't even mentioned the words "representation", "realization", "fundamental representation" etc... I use just the common sense notion: "hey, I want to transform this vector field in spacetime and I use these matrices because of Lorentz symmetry".

But, I do not understand when someone writes:

$$\psi' = S[\Lambda] \psi, \tag{2}$$

and call this a spinor! Where are its indices?

Now, to elaborate more:

a wavefunction is a scalar field, period. A pauli spinor and generally a Dirac spinor cannot be scalar fields, and therefore cannot be simple wave functions. So my question is:

since from non-relativistic quantum mechanics a Pauli spinor is: $$\big(\langle \vec{r}| \otimes \langle s|\big) |\psi\rangle := \psi(\vec{r},s) \equiv \begin{pmatrix}\psi(\vec{r},\uparrow) \\ \psi(\vec{r},\downarrow)\end{pmatrix} \equiv \begin{pmatrix}\psi(\vec{r},1) \\ \psi(\vec{r},2)\end{pmatrix} := \psi^{a}(\vec{r})$$ with $$a={1,2}$$, the index $$a$$ is from the lie algebra or from the spin? In other words, are the lie algebra indices synomys to spinor indices?

• Are the quote blocks from a reference? Which pages? Apr 7, 2023 at 10:20
• The first one is just to explain how I'm struggle with spinors. The second one is from "Quantum Mechanics - A.F.R de Toledo Piza, page 469" unfortunately is available just in portuguese. Apr 7, 2023 at 10:24

Let's take a step back: There are two ways to do linear algebra - in components and abstractly. For various reasons, physics texts often only do it in components (and this is the reason the question "where are its indices?" appears here).

When you have a vector space $$V$$ and a linear operator $$A : V\to V$$ on that vector space, then you can either write $$Av$$ to denote the action of $$A$$ on a vector $$v\in V$$, or you can choose a basis $$e_i \in V$$ and write the action of $$A$$ on a vector $$v$$ in terms of the matrix components as $$(Av)^i = {A^i}_j v^j$$.

Eq. (1) in the question is written in components, eq. (2) is not. They are equally statements about the action of certain linear operators on a vector space.

When we talk about non-relativistic spinors, we're talking about certain representations of the rotation algebra $$\mathfrak{su}(2)$$, i.e. we have some vector spaces $$V_\rho$$ with representation maps $$\rho : \mathfrak{su}(2)\to \mathfrak{gl}(V_\rho)$$, where $$\mathfrak{gl}(V_\rho)$$ is just the algebra of all linear operators on $$V_\rho$$.

For any such representation, the action of an infinitesimal rotation $$\omega\in\mathfrak{su}(2)$$ on a vector $$v\in V_\rho$$ can be written as either $$v' = \rho(\omega) v \tag{abstract}$$ or $$v'^i = {\rho(\omega)^i}_j v^j, \tag{components}$$ regardless of what representation it is - the indices come from a basis of $$V_\rho$$ and run from $$1$$ to $$\dim(V_\rho)$$.

It is customary in physics to use Greek indices when $$V_\rho$$ is Minkowski space, Latin indices starting from $$i$$ when $$V_\rho$$ is Euclidean space, and Latin indices starting from $$a$$ when $$V_\rho$$ is something else.

Non-relativistic spinors have a two-dimensional $$V_\rho$$, and so this falls into the "something else" category and the component notation will denote these spinors with $$\psi^a$$ in this convention.

Crucially, you have to distinguish this from an index on the Lie algebra $$\mathfrak{su}(2)$$ - as a Lie algebra, this is also a vector space, and you will see people talking about a basis $$T^a$$ of this vector space. Just because both indices are denoted with $$a$$ in these different contexts does not mean that this index is "the same" as that on the spinor $$\psi^a$$. The spinor index runs from 1 to 2, the index on the generators $$T^a$$ from 1 to 3.

• So in Pauli equation, $H\psi = i\partial_{t}\psi$, how can I write it using indices? And where lies the lie algebra indices, since the $a$ will be the spinor ones? Apr 7, 2023 at 17:10
• Because, my reasoning with spinor indices tends to write $\{\frac{1}{2m}[(-i\nabla_{i}-eA_{j})^2-e\sigma^{j}_{ab}B_{j}]+\delta_{ab}e\phi\}\psi^{a} = i\partial_{t}\psi^{b}$. But where are the lie algebra indices? Apr 7, 2023 at 17:13
• And the reason why we use $SU(2)$, a mathematical argument at least, is because $SO(3)$ doesn't have spinor representation? Therefore we are forced to represent it's double cover SU(2)? Apr 7, 2023 at 17:21
• @BasicMathGuy 1. I'm not 100% sure what you mean by a "Lie algebra index" - what do you think the $j$ on your $\sigma^j_{ab}$ is? Why do you think there should be "Lie algebra indices" in this equation? 2. See this answer of mine for why the universal cover appears in quantum mechanics. Apr 7, 2023 at 17:35
• Hey, I guess you're write....$j$ runs from $1,2,3$ which generates the whole thing. But It brings a confunsion for me too, since in gamma matrices $\gamma^{\mu}_{ab}$ the $\mu$ are the spacetime indices, but on its non-relativistic counterpart seems to be the $j$ in $\sigma^{j}_{ab}$ which are not spatial indices. Apr 7, 2023 at 17:58